Strategy11Handout - 10/9/2008 Mixed Strategies Matching...

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10/9/2008 1 Mixed Strategies – Matching Pennies ± In Matching Pennies, we found there was no Nash Equilibrium Guildenstern Heads Tails ± We expand the strategy set to allow for “mixed strategies” randomizations over Heads and Tales, the so-called “pure strategies” Rosencrantz Heads 1, -1 -1, 1 Tails -1, 1 1,-1 Why Mixed Strategies? ± If you were to play the same opponent repeatedly, you would like to be unpredictable . ± In rock-paper-scissors, if you behave predictably, your opponent will be able to beat you you. ± In a tennis match, you do not want your serves to be predicted by the opponent. Why Mixed Strategies? ± Maybe your opponent is picked from a large population of players… ± You play many times, each time with new opponent ± Each opponent does the same thing each time he plays ± Different opponents do different things ± You don’t know which type of opponent you get ± Model your opponent as randomly picking a strategy Why Mixed Strategies? ± Technical reason: Not all games have Nash equilibria in pure strategies. All games have Nash equilibria when allowing for mixed strategies. ± We are generalizing our notion of strategy ± pure strategies are a kind of mixed strategy ± where the probability distribution degenerates ± all the probability weight is on one pure strategy Mixed Strategies – Matching Pennies ± Rosencrantz’s strategy set is now a probability
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Strategy11Handout - 10/9/2008 Mixed Strategies Matching...

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